Theorem fvprif 13045

Description: The value of the pair function at an element of 2o. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
fvprif	|- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )

Proof of Theorem fvprif

Step	Hyp	Ref	Expression
1		fvpr0o 13043	. . . . 5 |- ( A e. V -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
2	1	3ad2ant1 1020	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
3	2	adantr 276	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
4		simpr 110	. . . 4 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> C = (/) )
5	4	fveq2d 5565	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) )
6	4	iftrued 3569	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> if ( C = (/) , A , B ) = A )
7	3, 5, 6	3eqtr4d 2239	. 2 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )
8		fvpr1o 13044	. . . . 5 |- ( B e. W -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
9	8	3ad2ant2 1021	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
10	9	adantr 276	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
11		simpr 110	. . . 4 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> C = 1o )
12	11	fveq2d 5565	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) )
13		1n0 6499	. . . . . 6 |- 1o =/= (/)
14	13	neii 2369	. . . . 5 |- -. 1o = (/)
15	11	eqeq1d 2205	. . . . 5 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( C = (/) <-> 1o = (/) ) )
16	14, 15	mtbiri 676	. . . 4 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> -. C = (/) )
17	16	iffalsed 3572	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> if ( C = (/) , A , B ) = B )
18	10, 12, 17	3eqtr4d 2239	. 2 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )
19		elpri 3646	. . . 4 |- ( C e. { (/) , 1o } -> ( C = (/) \/ C = 1o ) )
20		df2o3 6497	. . . 4 |- 2o = { (/) , 1o }
21	19, 20	eleq2s 2291	. . 3 |- ( C e. 2o -> ( C = (/) \/ C = 1o ) )
22	21	3ad2ant3 1022	. 2 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( C = (/) \/ C = 1o ) )
23	7, 18, 22	mpjaodan 799	1 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   (/)c0 3451   ifcif 3562   {cpr 3624   <.cop 3626   ` cfv 5259   1oc1o 6476   2oc2o 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-id 4329  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-res 4676  df-iota 5220  df-fun 5261  df-fv 5267  df-1o 6483  df-2o 6484

This theorem is referenced by: (None)